Yes. The proof that the canonical map is a bijection is constructive.
I have been asked to say more, which I do gladly. In constructive mathematics it is impossible to exhibit a discontinuous map (for instance, to define a discontinuous real function one needs excluded middle). We can even assume that all functions are continuous without getting into trouble, but we need not do that here.
In the situation at hand we proceed as follows. First we note that there is a model of constructive mathematics which includes the topological spaces. For instance filter spaces, limit spaces, or equilogical spaces -- almost any locally cartesian category extending topological spaces will do. For the argument at hand a cartesian closed category is probably enough (and this is what Alexander is doing when he cuts down to CGWH spaces, which I decipher as "Compactly generated Weak Hausdorff", that's a ccc). Thus, if we make an argument about topological spaces which is constructive, we can interpet it in this model to obtain a corresponding fact about topological spaces.
Inside the model everything looks like a set. We can thus treat pullbacks as in the category of sets, and there the canonical map
$$P^T \to A^T \times_{B^T} C^T$$
is easily seen to be a bijection, constructively (actually I'd be curious to see a natural non-constructive proof). We can write down the inverse explicitly. Thus we get two maps, which interpreted in our model yield continuous bijections.
The trouble with this method is that one has to learn to think constructively and to switch between external (outside the model) and internal (inside the model) reasoning. This is a big upfront cost, but once you've paid for it, you can pretend a lot of the time that everything is a set (when in reality it is a space, or a sheaf, or some such).
